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5.3 Adding and Subtracting Radical Expressions

Learning Objectives

Add and subtract like radicals.

Simplify radical expressions involving like radicals.

Adding and Subtracting Like Radicals

Adding and subtracting radical expressions is similar to adding and subtracting like terms. Radicals are considered to be like radicalsRadicals that share the same index and radicand., or similar radicalsTerm used when referring to like radicals., when they share the same index and radicand. For example, the terms 26 and 56 contain like radicals and can be added using the distributive property as follows:

26+56=(2+5)6=76

Typically, we do not show the step involving the distributive property and simply write,

26+56=76

When adding terms with like radicals, add only the coefficients; the radical part remains the same.

Example 1

Add: 753+353.

Solution:

The terms are like radicals; therefore, add the coefficients.

753+353=1053

Answer: 1053

Subtraction is performed in a similar manner.

Example 2

Subtract: 410−510.

Solution:

410−510=(4−5)10=−110=−10

Answer: −10

If the radicand and the index are not exactly the same, then the radicals are not similar and we cannot combine them.

Example 3

Simplify: 105+62−95−72.

Solution:

105+62−95−72=105−95+62−72=5−2

We cannot simplify any further because 5 and 2 are not like radicals; the radicands are not the same.

Answer: 5−2

Caution: It is important to point out that 5−2≠5−2. We can verify this by calculating the value of each side with a calculator.

5−2≈0.825−2=3≈1.73

In general, note that an±bn≠a±bn.

Example 4

Simplify: 5103+310−103−210.

Solution:

5103+310−103−210=5103−103+310−210=4103+10

We cannot simplify any further, because 103 and 10 are not like radicals; the indices are not the same.

Answer: 4103+10

Adding and Subtracting Radical Expressions

Often, we will have to simplify before we can identify the like radicals within the terms.

Example 5

Subtract: 32−18+50.

Solution:

At first glance, the radicals do not appear to be similar. However, after simplifying completely, we will see that we can combine them.

Similarly we can calculate the distance between (−3, 6) and (2,1) and find that c=52 units. Therefore, we can calculate the perimeter as follows:

P=a+b+c=52+25+52=102+25

Answer: 102+25 units

Key Takeaways

Add and subtract terms that contain like radicals just as you do like terms. If the index and radicand are exactly the same, then the radicals are similar and can be combined. This involves adding or subtracting only the coefficients; the radical part remains the same.

Calculate the perimeters of the triangles formed by the following sets of vertices.

{(−4, −5), (−4, 3), (2, 3)}

{(−1, 1), (3, 1), (3, −2)}

{(−3, 1), (−3, 5), (1, 5)}

{(−3, −1), (−3, 7), (1, −1)}

{(0,0), (2,4), (−2,6)}

{(−5,−2), (−3,0), (1,−6)}

A square garden that is 10 feet on each side is to be fenced in. In addition, the space is to be partitioned in half using a fence along its diagonal. How much fencing is needed to do this? (Round to the nearest tenth of a foot.)

A garden in the shape of a square has an area of 150 square feet. How much fencing is needed to fence it in? (Hint: The length of each side of a square is equal to the square root of the area. Round to the nearest tenth of a foot.)

Part C: Discussion Board

Choose values for x and y and use a calculator to show that x+y≠x+y.

Choose values for x and y and use a calculator to show that x2+y2≠x+y.